3.2 CAPM, Beta, and Cost of Equity

Building the Cost of Equity with CAPM

The cost of equity (Re) is the return common shareholders require given the risk of the residual claim. CFI's formula, used throughout FMVA valuation courses, is the capital asset pricing model (CAPM):

Re = Rf + β × (Rm − Rf)

Three pieces, each with a specific job:

1. Rf — the risk-free rate. In U.S. dollar DCFs, Rf is typically the yield on the 10-year U.S. Treasury note, matching a long-horizon cash-flow forecast. Rf is an observable market yield on the date you build the model. It is not a CFI constant and not an official exam number. Do not memorize a live Treasury print from a blog, a prior course slide, or this guide as if it were "the" Rf. On the exam, the vignette will give Rf or tell you which Treasury yield to take. On a live model, you look up the 10-year (or a matching-currency long government yield) and you date-stamp it.

2. β — equity (levered) beta. Beta measures how the stock's excess return moves with the market's excess return. β = 1.0 means the equity moves with the market; β > 1.0 means it amplifies market moves; β < 1.0 means it dampens them. The beta that belongs in CAPM for the firm's common stock is levered beta, because shareholders bear both asset risk and financial leverage.

3. ERP = Rm − Rf — the equity risk premium. This is the extra return investors demand for holding a diversified equity portfolio rather than Treasuries. Banks and valuation groups often take ERP from Kroll / Morningstar publications (the series formerly associated with Duff & Phelps) rather than estimating Rm from a short recent bull or bear market. A two-year realized S&P return is not an ERP. Mixing a 20% trailing-year market gain into CAPM as if it were Rm produces a nonsense Re that will not survive a review.

Worked CAPM Example

A case study supplies Rf = 4.0% as the 10-year Treasury yield on the valuation date, levered beta βL = 1.15, and ERP = 5.0% from the Kroll/Morningstar source the desk uses.

Re = 4.0% + 1.15 × 5.0% = 4.0% + 5.75% = 9.75%

If you mistakenly use a 1-year T-bill of 5.2% as Rf with a long-term ERP of 5.0%, you mix a short rate with a long premium and overstate Re. Match the horizon of Rf to the horizon of the cash flows. If the DCF is a 10-year explicit forecast plus a terminal value, a 10-year Treasury is the standard U.S. choice. A 3-month bill belongs next to a 3-month cash account, not next to a going-concern enterprise valuation.

If ERP is 4.5% instead of 5.0% with the same Rf and beta, Re = 4.0% + 1.15 × 4.5% = 9.175%. A 50-basis-point ERP choice moves Re by 1.15 × 50 bp = 58 bp. That is why ERP is a documented research input, not a cell you type from memory of last year's market.

Levered Beta versus Unlevered (Asset) Beta

Observed betas from Bloomberg, Capital IQ, or a regression on the company's own stock are levered betas: they embed the company's current or historical debt. Two firms with identical operations can show different equity betas solely because one is more levered.

CFI's unlever / relever pair (the Hamada-style identities taught in the core):

Unlevered beta = Levered beta / (1 + (1 − t) × (D/E))

Levered beta = Unlevered beta × (1 + (1 − t) × (D/E))

Unlevered beta (also called asset beta) strips out financial leverage so you can compare operating risk across peers. Relevered beta puts the peer-average operating risk onto the target D/E of the firm you are valuing. The tax rate t in these formulas is the same marginal rate you will use as T in WACC; using 21% to unlever and 25% in WACC is an avoidable inconsistency.

Worked Unlever / Relever Example

You are valuing Northline Logistics. You will use a 25% target debt-to-value, so D/E = 0.25 / 0.75 = 0.333, and tax rate t = 25%.

A peer, SwiftHaul, has a levered beta of 1.40, market D/E of 0.80, and t = 25%.

βu = 1.40 / (1 + (1 − 0.25) × 0.80) = 1.40 / (1 + 0.75 × 0.80) = 1.40 / (1 + 0.60) = 1.40 / 1.60 = 0.875

Relever to Northline's target:

βL = 0.875 × (1 + (1 − 0.25) × 0.333) = 0.875 × (1 + 0.75 × 0.333) = 0.875 × (1 + 0.250) = 0.875 × 1.250 = 1.094

If Rf = 4.0% and ERP = 5.0%, Northline Re = 4.0% + 1.094 × 5.0% = 9.47%.

If you had plugged SwiftHaul's raw 1.40 into Northline's CAPM, Re would be 4.0% + 7.0% = 11.0% — too high, because you imported SwiftHaul's heavier leverage. That 153-basis-point gap is entirely a capital-structure error, not an operating-risk difference.

Check the algebra the other way. Relever 0.875 at SwiftHaul's own D/E of 0.80 and you must recover 1.40: 0.875 × (1 + 0.75 × 0.80) = 0.875 × 1.60 = 1.40. If you cannot round-trip the peer, you have the wrong t or the wrong D/E in the unlever step.

Peer Betas, Excel SLOPE, and Pairing Traps

The Peer-Beta Method

Private companies and thinly traded public companies do not have a usable stock-price history. Even liquid names can have noisy two-year betas. The peer-beta method is the FMVA-standard workaround:

1. Select a peer set with similar operating risk (same industry, similar margins and cyclicality). Do not pick a peer only because it is in the same index.
2. Collect each peer's levered beta, market D/E, and marginal tax rate.
3. Unlever each peer beta to asset beta with CFI's formula.
4. Average or take the median of the unlevered betas. Medians are more robust to one outlier high-beta name.
5. Relever the average unlevered beta at the subject firm's target D/E and tax rate — the same target mix that will enter WACC.
6. Insert that relevered beta into CAPM with the case's Rf and ERP.

Worked three-peer set (all t = 25%):

Mean βu = (0.875 + 0.846 + 0.826) / 3 = 0.849. Relever at Northline's target D/E of 0.333: βL = 0.849 × 1.250 = 1.061. Re = 4.0% + 1.061 × 5.0% = 9.31%.

Do not average the raw levered betas (1.40, 1.10, 0.95) → 1.15 and drop 1.15 into Northline's CAPM. That silently blends three different financing policies. Unlever first.

Estimating Beta in Excel with SLOPE

When you do have a price history, CFI-style Excel work uses a regression of stock excess returns on market excess returns. A direct worksheet function is SLOPE:

=SLOPE(stock_excess_returns, market_excess_returns)

If you work with total returns rather than excess returns and Rf is stable over the window, SLOPE of stock returns on market returns is a close approximation to beta. Weekly returns over 2 years (about 104 points) or monthly returns over 5 years (60 points) are common windows. Practical construction:

• Pull adjusted close prices for the stock and for a market proxy (S&P 500 for a U.S. firm) so dividends are in the return.
• Compute periodic returns: =P1/P0 - 1 (or =LN(P1/P0) if the case specifies log returns; do not mix the two in one regression).
• Optionally subtract the matching-period Rf to get excess returns.
• =SLOPE(stock_range, market_range) is β.
• =INTERCEPT(...) is alpha; =RSQ(...) tells you how much of the stock's variance the market explained. A very low R² means the beta is noisy — another reason to prefer a peer median.

Traps in the regression: using raw price levels instead of returns (SLOPE on prices is not beta); mixing daily stock returns with weekly market returns; forgetting a dividend-adjusted index when the stock pays a large yield; using a 30-day window that is just one news event. Book value never enters the SLOPE range. There is no "book beta."

Exam Traps CFI Wants You to Catch

Trap 1: "Book beta." Beta is estimated from market returns, not from book-value ratios. Using book D/E in the unlever formula when market D/E is available is also inconsistent with market-value WACC. Book D/E is a leverage ratio; it is not a beta.

Trap 2: Mixing levered beta with unlevered cash flows. UFCF is cash flow to all providers of capital, before interest. The matching discount rate is WACC, which already blends Re (from levered beta) with after-tax Rd. If you discount UFCF at Re, you put leverage into the rate without giving debt its claim on the cash flow — you will misstate enterprise value. Conversely, discounting free cash flow to equity (FCFE) at WACC is the opposite mismatch. Pairing that the exam expects: UFCF ↔ WACC ↔ enterprise value; FCFE ↔ Re ↔ equity value.

Trap 3: Using a peer's levered beta without unlevering. You import the peer's financing policy into a firm that may have a different target mix.

Trap 4: Relevering at current D/E when you will use target weights in WACC. Beta and WACC weights must tell the same leverage story. Harbor Tools' target 30% D/V (D/E = 0.429) is the relever ratio if that is the WACC mix; the current 23.1% D/V is not.

Trap 5: Treating ERP as last year's stock-market return. ERP is a forward long-run premium, often taken from Kroll/Morningstar publications.

Trap 6: Inventing Rf. On a live model you look up the 10-year Treasury. On the exam you use the rate the vignette gives. There is no single official Rf baked into FMVA, and this guide will not pretend there is.

When beta, Rf, and ERP are internally consistent with the target mix, Re is ready to drop into WACC.